Numerical schemes for kinetic equation with diffusion limit and anomalous time scale

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In this work, we propose numerical schemes for linear kinetic equation, which are able to deal with a diffusion limit and an anomalous time scale of the form \begin{document}${\varepsilon ^2}\left( {1 + \left| {\ln \left( \varepsilon \right)} \right|} \right)$\end{document} . When the equilibrium distribution function is a heavy-tailed function, it is known that for an appropriate time scale, the mean-free-path limit leads either to diffusion or fractional diffusion equation, depending on the tail of the equilibrium. This work deals with a critical exponent between these two cases, for which an anomalous time scale must be used to find a standard diffusion limit. Our aim is to develop numerical schemes which work for the different regimes, with no restriction on the numerical parameters. Indeed, the degeneracy \begin{document}$ \varepsilon\to0$\end{document} makes the kinetic equation stiff. From a numerical point of view, it is necessary to construct schemes able to undertake this stiffness to avoid the increase of computational cost. In this case, it is crucial to capture numerically the effects of the large velocities of the heavy-tailed equilibrium. Moreover, we prove that the convergence towards the diffusion limit happens with two scales, the second being very slow. The schemes we propose are designed to respect this asymptotic behavior. Various numerical tests are performed to illustrate the efficiency of our methods in this context.